Analyzing a RCBD
Treatment Structure: One-way with recipe (3-levels: classic butter, brown butter, coconut oil)
Design Structure: Recipe randomly assigned to tray (e.u.) in an RCBD with r = 4 ovens (blocking factor). Texture recorded for each tray (m.u.).
# A tibble: 12 × 3
oven recipe texture
<chr> <chr> <dbl>
1 A Classic Butter 80.6
2 A Brown Butter 84.3
3 A Coconut Oil 78.5
4 B Classic Butter 59.9
5 B Brown Butter 66.4
6 B Coconut Oil 60.7
7 C Classic Butter 75.1
8 C Brown Butter 79.9
9 C Coconut Oil 74.6
10 D Classic Butter 62.5
11 D Brown Butter 65.0
12 D Coconut Oil 60.9
If we analyze ignoring the effect of oven…
Incorrect Analysis: CRD
\(y_{ij} = \mu + \tau_i + \epsilon_{ij} \text{ where } \epsilon_{ij} \text{iid}\sim N(0,\sigma^2)\)
\[y_{ij} = \mu + \tau_i + \rho_j + \epsilon_{ij} \text{ where } \epsilon_{ij} \text{iid}\sim N(0,\sigma^2)\] for \(i = 1, 2, 3\) and \(j = 1, 2, 3, 4\)
where:
\[SST = SSTrt + SSBlk + SSE\]
| Source of Variation | DF | SS | MS | F |
|---|---|---|---|---|
| Block | r - 1 | SSBlk | MSBlk | MSBlk/MSE |
| Treatment | t - 1 | SSTrt | MSTrt | MSTrt/MSE |
| Block x Treatment \(\rightarrow\) error | (r - 1)(t - 1) | SSE | MSE \(\rightarrow \sigma^2\) | |
| Total (N = rt) | N - 1 | SST |
Blocking is a design tool, not usually a research question.
| Source of Variation | DF |
|---|---|
Analysis of Variance Table
Response: texture
Df Sum Sq Mean Sq F value Pr(>F)
recipe 2 63.82 31.910 38.607 0.0003749 ***
oven 3 825.28 275.092 332.829 4.651e-07 ***
Residuals 6 4.96 0.827
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
recipe emmean SE df lower.CL upper.CL .group
Brown Butter 73.9 0.455 6 72.4 75.4 A
Classic Butter 69.5 0.455 6 68.0 71.0 B
Coconut Oil 68.7 0.455 6 67.2 70.2 B
Results are averaged over the levels of: oven
Confidence level used: 0.95
Conf-level adjustment: sidak method for 3 estimates
P value adjustment: tukey method for comparing a family of 3 estimates
significance level used: alpha = 0.05
NOTE: If two or more means share the same grouping symbol,
then we cannot show them to be different.
But we also did not show them to be the same.
If we ignore ovens (CRD):
Analysis of Variance Table
Response: texture
Df Sum Sq Mean Sq F value Pr(>F)
recipe 2 63.82 31.910 0.3459 0.7166
Residuals 9 830.24 92.248
With RCBD:
Analysis of Variance Table
Response: texture
Df Sum Sq Mean Sq F value Pr(>F)
recipe 2 63.82 31.910 38.607 0.0003749 ***
oven 3 825.28 275.092 332.829 4.651e-07 ***
Residuals 6 4.96 0.827
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Recall: The experimental error comes from the block x treatment term.